1+1=2
- Differential equation
- Gaussian integral (integral from -inf to +inf of exp(-x²) = sqrt(pi))
- Cauchy–Riemann equations
- Chapman–Kolmogorov equation
- Clairaut's equation
- Fredholm integral equation
- Hill differential equation
- Ishimori equation
- Laplace's equation
- Maurer–Cartan equation
- Pell's equation
- Poisson's equation
- Riccati equation
- Sine–Gordon equation
- Verhulst equation
What an interesting question. However it is ill-defined what is "most important". The question asks for the most important formulas in mathematics. So, I guess any physical formula would already be "too applied". The problem is that mathematical formulas too are dwarfs on the shoulders of giants. If you seek these "giants", then you will find yourself at the historical roots of mathematics and probably end up with the Pythagorean theorem (already mentioned) as a candidate for the most important formula, or some set of axioms. On the far other end, maybe you think about a very common, yet specific application in computation science and might suggest something like the set of formulas which describe the fast Fourier transformation.
I personally think, the standard answer which should come to mind, when being asked what the most important mathematical formula is, should be somehow the following: The most powerful mathematical tool is calculus. Out of all mathematics it is already so advanced, that you do not take it subconsciencely for granted, like the Pythagorean theorem. (Basicly our school life is devided into pre-calculus and calculus.) But on the other hand it is still general enough that it is no exaggeration to say that it is one of the most powerful tool in mathematics and natural sciences. Therefore the most important formulas should be something like the definition of the first derivative as a limit paired together with the fundamental theorem of calculus (that the area under a curve turns out to be the inverse of finding the slope of a curve).
One could maybe mention that there is the generalised Stokes Theorem being basicly the n-dimensional generalisation of the fundamental theorem of calculus, and it looks even simpler and thus a lot more beautiful, but to be honest, I did not actually needed it in exactly that form so often.
I hope this answer was useful.
"Most important" - for what? For applications, for understanding our universe, for understanding our number system and system of logic, etc.
There are many answers. Most often mathematicians are interesting in characterizing to with in an equivalence class the characterizes that specify that class and deriving the invariants of that class. For example, the concept of congruence in plane geometry gives a characterization of all triangles in various ways.
The Atiyah-Singer Index Theorem which gives the equality of the analytic index (related to the space of solutions of an elliptic differential operator) to the topological index on a compact manifold. The there is Bezout's Theorem - a formula of the number of points of intersection of two algebraic varieties over a algebraically closed field. The Cauchy-Reimann equations gives a formula that specifies holomorphic functions. The De Rahm Cohomology is a classic formula for classified the topological properties of C-infinity manifolds and the Dolbeault Cohomology expands that to complex holomorphic manifolds.
This question is a broad as all of mathematics and of course the answer will vary depending on a lot of factors.
Interesting if not important formulae:
exp(i.pi)+1=0
sum of reciprocals of sqares of all positive integers =(sqare of pi)/6.
v+f-e=2
In graduate school I took a year of several complex variables and a year of algebraic geometry under Wei-Liang Chow.
https://en.wikipedia.org/wiki/Wei-Liang_Chow
Chow, Oscar Zariski and Andre Weil are credited for wrapping a logical and algebraic foundation around the intuition of the Italian school of algebraic geometry (while their logic and proofs left something to be desired) and creating the foundations modern algebraic geometry as an important area of mathematics by developing an equivalent dictionary between commutative algebra and the geometry of algebraic sets over fields. Of course that was expanded by Serre and and Grothendieck to look at algebraic sets over rings which which lead to powerful tools in both geometry and number theory.
Anyway Chow was a night owl and you could often find him in his office at 10 PM. We would often have free ranging conversations late into the night. That man always had a kind word and did he loved to talk. One time I asked him what was the single greatest advance in mathematics.
He thought a minute and said - the complex differentiation of a function of a complex variable and recognizing the implications. To handle multivalued functions of a complex variable like sqrt( z), log(z) where z is complex let to the definition of a Riemann surface and to general manifolds. That sounds trivial but - it opened up an extraordinarily deep and beautiful area of mathematics that has applications far and wide from number theory to geometry to physics. The current best model if the extra dimensions needed for string theory are Calabi-Yau manifolds which are very special complex analytic manifolds.
So the poster's question has many answers I suspect but it got me thinking about that conversation with Chow many years ago.
There are many, many answers to a question such as (quote): 'What are some of the most important formulas in mathematics?'. If you look at the responses provided by Profs/Drs/Mr/Eng(s) Jochen Wilhelm, Patrick Poschke, Truman Prevatt and (interestingly enough) Narendra Ladhawala as well as the others there is a 'very good range of most important formulas in mathematics' that have been presented. However, it is really a case of 'which of those most important formulae in mathematics' suits your needs and most (your most) commonly used applications. As a specialist in dam engineering, dam safety and risk assessment for dams, I would use many of those listed BUT very rarely most of them. So, whether scientist, engineer, researcher, student or one who is interested in what is called 'recreational mathematics' the range of 'most important formulas in mathematics' is very long.
all mathematical fields are important, but i see that numerical formulas are the best because these formulas helps us to find a solution ( approximate solution)for For all problems that can not be solved by analytical methods
All mathematical formulas are important to their duties. It depends on what one wants to solve...
I am writing this, because I got asked recently, refering to this discussion, if I think mathematical formulas are important. Here is my answer to that question and you may quote me, if you want:
Yes, of course, mathematical formulas are important! They often can express a calculation much more precise than any human language could with relatively few symbols. This is especially true, if the calculation has reached a certain level of complexity. This becomes very obvious, when you look at the mathematical results of, say Galilei, or many ancient Greeks. They had to formulate basic geometric theorems as very long paragraphs. They are hard grasp. Nowadays, we would just draw a good sketch and write one line of mathematical symbols. We just have to learn once, what they mean. Then we can look at a simple formula and almost immediately understand what is going on.
Even better, since there are many conventions for symbols, many scientists often can instantly recognise many basic simple formulas, even without context. But not only in everyday life are formulas important to help you grasp simple calculations like transforming between different measurement units. Also in science they are very important. I claim that many parts of mathematics and natural sciences would not even work the way we are used to it without mathematical formulas. And I think most mathematicians and natural scientists will agree with me on that.
Furthermore, an often forgotten advantage of formulas is the simple fact that due to the rules of equivalent transformations, one single formula is not simply one statement, but several at once. Because of these rules, all other forms you can put the formula in are implied as well. Try that with a complicated phrase! Whenever there is a double negation in a sentence, you ask yourself, if you can simply cross them out, because between two extremes there is often still the third, neutral state. There you can see that one can easily calculate with formulas - not with phrases.In formulas one has a high information density and almost no superfluous, grammatical ballast of many languages.
I hope that I could convince you that mathematical formulas are very important. Sure, you don't have to memorise all formulas. They can be looked up. However, if you remember all important formulas of a topic, you can calculate much faster. Furthermore, one can often simply remember the most basic formulas exactly, the other ones just approximately and then derive the exact forms of all the formulas once you understood them fully, because they are all connected.
Have a nice day, everyone.
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